Meromorphic solutions of delay differential equations related to logistic type and generalizations

Abstract

Let {bj}j=1k be meromorphic functions, and let w be admissible meromorphic solutions of delay differential equationw′(z)=w(z)[P(z,w(z))Q(z,w(z))+∑j=1kbj(z)w(z−cj)] with distinct delays c1,…,ck∈C∖{0}, where the two nonzero polynomials P(z,w(z)) and Q(z,w(z)) in w with meromorphic coefficients are prime each other. We obtain that if limsupr→∞log⁡T(r,w)r=0, thendegw⁡(P/Q)≤k+2. Furthermore, if Q(z,w(z)) has at least one nonzero root, then degw⁡(P)=degw⁡(Q)+1≤k+2; if all roots of Q(z,w(z)) are nonzero, then degw⁡(P)=degw⁡(Q)+1≤k+1; if degw⁡(Q)=0, then degw⁡(P)≤1.In particular, whenever degw⁡(Q)=0 and degw⁡(P)≤1 and without the growth condition, any admissible meromorphic solution of the above delay differential equation (called Lenhart-Travis' type logistic delay differential equation) with reduced form can not be an entire function w satisfying N‾(r,1w)=O(N(r,1w)); while if all coefficients are rational functions, then the condition N‾(r,1w)=O(N(r,1w)) can be omitted. Furthermore, any admissible meromorphic solution of the logistic delay differential equation (that is, for the simplest special case where k=1 and degw⁡(P/Q)=0) satisfies that N(r,w) and T(r,w) have the same growth category. Some examples support our results.